Boundary-Layer Meteorology Nondimensionalization of A Turbulent Boundary–Layer System: Obukhov Length and Monin-Obukhov Similarity Theory Jun-Ichi Yano, Marta Waclawczyk To cite this version: Jun-Ichi Yano, Marta Waclawczyk. Boundary-Layer Meteorology Nondimensionalization of A Turbu- lent Boundary–Layer System: Obukhov Length and Monin-Obukhov Similarity Theory. Boundary- Layer Meteorology, Springer Verlag, In press. hal-03275540 HAL Id: hal-03275540 https://hal.archives-ouvertes.fr/hal-03275540 Submitted on 1 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. 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Boundary-Layer Meteorology Nondimensionalization of A Turbulent Boundary--Layer System: Obukhov Length and Monin-Obukhov Similarity Theory --Manuscript Draft-- Manuscript Number: BOUN-D-21-00049R2 Full Title: Nondimensionalization of A Turbulent Boundary--Layer System: Obukhov Length and Monin-Obukhov Similarity Theory Article Type: Research Article Keywords: Nondimensionalization scale; Obukhov length; Stably-stratified turbulence; Vertical scale; TKE equation Corresponding Author: jun-ichi yano Meteo France FRANCE Corresponding Author Secondary Information: Corresponding Author's Institution: Meteo France Corresponding Author's Secondary Institution: First Author: jun-ichi yano First Author Secondary Information: Order of Authors: jun-ichi yano Marta Waclawczyk Order of Authors Secondary Information: Funding Information: Narodowe Centrum Nauki Dr Marta Waclawczyk (2020/37/B/ST10/03695) Abstract: The Obukhov length, although often adopted as a characteristic scale of atmospheric boundary layers, has been introduced purely based on a dimensional argument without a deductive derivation from a governing equation system. Here, its derivation is pursued by the nondimensionalization method in the same manner as for the Rossby deformation radius and the Ekman-layer depth. Physical implications of the Obukhov length is inferred by nondimensionalizing the turbulent kinetic energy (TKE) equation for the horizontally-homogeneous boundary layer. A nondimensionalization scale for a full set of equations for the boundary-layer eddies formally reduces to the Obukhov length by multiplying a re-scaling factor to the former. This re-scaling factor increases with increasing stable stratifications of the boundary layer, in which flows tend to be more horizontal and gentler, thus the Obukhov length increasingly loses its relevance. A heuristic, but deductive derivation of the Monin- Obukhov similarity theory is also outlined based on the obtained nondimensionalization results. Response to Reviewers: please refer to the uploaded file (reply.pdf) Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation Manuscript Click here to access/download;Manuscript;ms.pdf Click here to view linked References Noname manuscript No. (will be inserted by the editor) 1 Nondimensionalization of A Turbulent 2 Boundary–Layer System: Obukhov Length and 3 Monin–Obukhov Similarity Theory 4 Jun–Ichi Yano · Marta Wac lawczyk 5 6 Received: DD Month YEAR / Accepted: DD Month YEAR 7 DOC/PBL/Monin-Obukhov/BLM/ms.tex, June 28, 2021 8 Abstract The Obukhov length, although often adopted as a characteris- 9 tic scale of atmospheric boundary layers, has been introduced purely based 10 on a dimensional argument without a deductive derivation from a governing 11 equation system. Here, its derivation is pursued by the nondimensionalization 12 method in the same manner as for the Rossby deformation radius and the 13 Ekman–layer depth. Physical implications of the Obukhov length is inferred 14 by nondimensionalizing the turbulent kinetic energy (TKE) equation for the 15 horizontally-homogeneous boundary layer. A nondimensionalization scale for 16 a full set of equations for the boundary–layer eddies formally reduces to the 17 Obukhov length by multiplying a re–scaling factor to the former. This re– 18 scaling factor increases with increasing stable stratifications of the boundary 19 layer, in which flows tend to be more horizontal and gentler, thus the Obukhov 20 length increasingly loses its relevance. A heuristic, but deductive derivation of 21 the Monin–Obukhov similarity theory is also outlined based on the obtained 22 nondimensionalization results. 23 Keywords Nondimensionalization scale · Obukhov length · Stably–stratified 24 turbulence · Vertical scale · TKE equation · 25 1 Introduction 26 Theories of atmospheric boundary–layer turbulence have been developed by 27 heavily relying on the so–called dimensional analyses (Barenblatt 1996). This J.–I. Yano CNRM, UMR3589 (CNRS), M´et´eo-France, 31057 Toulouse Cedex, France E-mail: [email protected] M. Wac lawczyk Institute of Geophysics, Faculty of Physics, University of Warsaw, Warsaw, Poland E-mail: [email protected] 2 Jun–Ichi Yano, Marta Wac lawczyk 28 methodology is alternatively called scaling in atmospheric boundary–layer 29 studies, as reviewed by e.g., Holtslag and Nieuwstadt (1986), Foken (2006). 30 Some key variables controlling a given regime of turbulence are first identified, 31 then various characteristic scales of the system (e.g., length, velocity, temper- 32 ature) are determined from these key controlling variables by a dimensional 33 consistency. For example, the Obukhov length (Obukhov 1948) is defined, by 34 dimensional analysis, from the frictional velocity and the buoyancy flux. Re- 35 sulting theories from these analyses are called “similarity theories”, because 36 they remain similar regardless of specific cases by re–scaling the variables in 37 concern by characteristic scales. 38 In performing a dimensional analysis correctly, a certain ingenuity is re- 39 quired for choosing proper controlling variables of a given system. No system- 40 atic methodology exists for choosing them, but the choice is solely based on 41 physical intuitions. A wrong choice of controlling variables can lead to totally 42 meaningless results (cf., Batchelor 1954). With the absence of an analytical 43 solution to turbulent flows as well as difficulties in observations and numerical 44 modelling, usefulness of those proposed scales is often hard to judge. As a re- 45 sult, the Obukhov length is hardly a unique choice. There are various efforts to 46 introduce alternative scales, as further discussed in Sects. 2.4 and 6.1, but the 47 theory based on gradient–based scales (Sorbjan 2006, 2010, 2016) is probably 48 the most notable. 49 However, the dimensional analysis is not a sole possibility of defining char- 50 acteristic scales of a system. In atmospheric large-scale dynamics, these scales 51 are typically derived by nondimensionalizations. No doubt, this procedure is 52 more straightforward and formal: characteristic scales are introduced for all 53 the variables of a system for nondimensionalizing them. These scales cannot 54 be arbitrary, because we expect that terms in an equation balance each other, 55 thus their orders of magnitudes must match each other. These conditions, in 56 turn, constrain these characteristic scales in a natural manner. Advantage of 57 the nondimensionalization analysis is that these scales are defined not only 58 by dimensional consistencies, but also by requirements of balance by order of 59 magnitude between the terms in a system. The Rossby radius of deformation 60 is a classical example of a characteristic scale identified by a nondimensional- 61 ization. This scale characterizes the quasi-geostrophic system (Sect. 3.12, Ped- 62 losky 1987). The depth of the Ekman layer is another such example (Sect. 4.3, 63 Pedlosky 1987). Yano and Tsujimua (1987), Yano and Bonazzola (2009) sys- 64 tematically apply this methodology for the scale analysis. 65 In a certain sense, nondimensionalization is a brute approach without re- 66 lying on any physical intuitions, even observation or modeling. The procedure 67 is totally formal and abstract. However, it can lead us to identify physically 68 meaningful scales, because it is applied to equations that govern a given system 69 physically: see e.g., discussions concerning Eqs. (5a, b) in Sect. 2.3 below. 70 Note a subtle difference of the concept of the scale in nondimensionalization 71 from that in similarity theories: in similarity theories, precise values are used 72 to scale the variables in universal functions that define a solution of a system. 73 On the other hand, under the nondimensionalization analysis, the scales are Obukhov Length As a Nondimensionalization Scale 3 74 typically defined by orders of magnitudes, as just stated. This difference must 75 clearly be kept in mind in the following analyses: see further discussions in 76 Sects. 5. However, in principle, the characteristic scales adopted in boundary– 77 layer similarity theories should be linked to the nondimensionalization scales 78 identified by nondimensionalization analysis, if the former have physical basis 79 to be adopted. The most basic motivation of the present study is to investigate 80 whether this is the case. 81 As a first step of such a systematic investigation, the present study focuses 82 on the Obukhov